The potential we are investigating has the form
\begin{equation}
V(\phi) = c\phi + \frac{\phi^{2}R}{12} - \lambda\phi^{4}
\end{equation}
where $c$ is some constant term, so its first derivative would be
\begin{equation}
V'(\phi) = c + \frac{\phi R}{6} - 4\lambda\phi^3.
\end{equation}
By the fundamental theorem of algebra, it has exactly three roots at the values
\begin{eqnarray*}
v_{1} &=& -\frac{\sqrt[3]{2} R \lambda +\left(\sqrt{2}
    \sqrt{\lambda ^3 \left(5832 c^2 \lambda -R^3\right)}-108 c \lambda
    ^2\right)^{2/3}}{6 2^{2/3} \lambda  \sqrt[3]{\sqrt{2} \sqrt{\lambda ^3
    \left(5832 c^2 \lambda -R^3\right)}-108 c \lambda ^2}} \\
v_{2} &=& \frac{2 \sqrt[3]{-2} R \lambda +\left(1-i \sqrt{3}\right)
    \left(\sqrt{2} \sqrt{\lambda ^3 \left(5832 c^2 \lambda -R^3\right)}-108 c
    \lambda ^2\right)^{2/3}}{12 2^{2/3} \lambda  \sqrt[3]{\sqrt{2}
    \sqrt{\lambda ^3 \left(5832 c^2 \lambda -R^3\right)}-108 c \lambda
    ^2}} \\
v_{3} &=& \frac{\left(1+i \sqrt{3}\right)
    \left(\sqrt{2} \sqrt{\lambda ^3 \left(5832 c^2 \lambda -R^3\right)}-108 c
    \lambda ^2\right)^{2/3}-2 (-1)^{2/3} \sqrt[3]{2} R \lambda }{12 2^{2/3}
    \lambda  \sqrt[3]{\sqrt{2} \sqrt{\lambda ^3 \left(5832 c^2 \lambda
    -R^3\right)}-108 c \lambda ^2}}.
\end{eqnarray*}
We can plug each of these into the function with the extra linear term
and find
\begin{eqnarray*}
V(v_{1}) &=&    -\frac{\left(\sqrt[3]{2} R \lambda +\left(\sqrt{2} \sqrt{\lambda ^3
    \left(5832 c^2 \lambda -R^3\right)}-108 c \lambda
    ^2\right)^{2/3}\right)^4}{5184 2^{2/3} \lambda ^3 \left(\sqrt{2}
    \sqrt{\lambda ^3 \left(5832 c^2 \lambda -R^3\right)}-108 c \lambda
    ^2\right)^{4/3}}\\
& &+\frac{R \left(\sqrt[3]{2} R \lambda +\left(\sqrt{2}
    \sqrt{\lambda ^3 \left(5832 c^2 \lambda -R^3\right)}-108 c \lambda
    ^2\right)^{2/3}\right)^2}{864 \sqrt[3]{2} \lambda ^2 \left(\sqrt{2}
    \sqrt{\lambda ^3 \left(5832 c^2 \lambda -R^3\right)}-108 c \lambda
    ^2\right)^{2/3}} \\
& &-\frac{c \left(\sqrt[3]{2} R \lambda +\left(\sqrt{2}
    \sqrt{\lambda ^3 \left(5832 c^2 \lambda -R^3\right)}-108 c \lambda
    ^2\right)^{2/3}\right)}{6 2^{2/3} \lambda  \sqrt[3]{\sqrt{2} \sqrt{\lambda
    ^3 \left(5832 c^2 \lambda -R^3\right)}-108 c \lambda ^2}}
\end{eqnarray*}
\begin{eqnarray*}
V(v_2) &=& -\frac{\left(2 \sqrt[3]{-2} R \lambda +\left(1-i \sqrt{3}\right)
    \left(\sqrt{2} \sqrt{\lambda ^3 \left(5832 c^2 \lambda -R^3\right)}-108 c
    \lambda ^2\right)^{2/3}\right)^4}{82944 2^{2/3} \lambda ^3 \left(\sqrt{2}
    \sqrt{\lambda ^3 \left(5832 c^2 \lambda -R^3\right)}-108 c \lambda
    ^2\right)^{4/3}} \\
& & +\frac{R \left(2 \sqrt[3]{-2} R \lambda +\left(1-i
    \sqrt{3}\right) \left(\sqrt{2} \sqrt{\lambda ^3 \left(5832 c^2 \lambda
    -R^3\right)}-108 c \lambda ^2\right)^{2/3}\right)^2}{3456 \sqrt[3]{2}
    \lambda ^2 \left(\sqrt{2} \sqrt{\lambda ^3 \left(5832 c^2 \lambda
    -R^3\right)}-108 c \lambda ^2\right)^{2/3}} \\
& & +\frac{c \left(2 \sqrt[3]{-2} R
    \lambda +\left(1-i \sqrt{3}\right) \left(\sqrt{2} \sqrt{\lambda ^3
    \left(5832 c^2 \lambda -R^3\right)}-108 c \lambda
    ^2\right)^{2/3}\right)}{12 2^{2/3} \lambda  \sqrt[3]{\sqrt{2}
    \sqrt{\lambda ^3 \left(5832 c^2 \lambda -R^3\right)}-108 c \lambda ^2}}
\end{eqnarray*}
\begin{eqnarray*}
V(v_{3}) &=& -\frac{\left(\left(1+i \sqrt{3}\right) \left(\sqrt{2} \sqrt{\lambda ^3
    \left(5832 c^2 \lambda -R^3\right)}-108 c \lambda ^2\right)^{2/3}-2
    (-1)^{2/3} \sqrt[3]{2} R \lambda \right)^4}{82944 2^{2/3} \lambda ^3
    \left(\sqrt{2} \sqrt{\lambda ^3 \left(5832 c^2 \lambda -R^3\right)}-108 c
    \lambda ^2\right)^{4/3}} \\ 
& &+\frac{R \left(\left(1+i \sqrt{3}\right)
    \left(\sqrt{2} \sqrt{\lambda ^3 \left(5832 c^2 \lambda -R^3\right)}-108 c
    \lambda ^2\right)^{2/3}-2 (-1)^{2/3} \sqrt[3]{2} R \lambda \right)^2}{3456
    \sqrt[3]{2} \lambda ^2 \left(\sqrt{2} \sqrt{\lambda ^3 \left(5832 c^2
    \lambda -R^3\right)}-108 c \lambda ^2\right)^{2/3}} \\
& &+\frac{c
    \left(\left(1+i \sqrt{3}\right) \left(\sqrt{2} \sqrt{\lambda ^3 \left(5832
    c^2 \lambda -R^3\right)}-108 c \lambda ^2\right)^{2/3}-2 (-1)^{2/3}
    \sqrt[3]{2} R \lambda \right)}{12 2^{2/3} \lambda  \sqrt[3]{\sqrt{2}
    \sqrt{\lambda ^3 \left(5832 c^2 \lambda -R^3\right)}-108 c \lambda ^2}}.
\end{eqnarray*}
Observe that these are nonzero quantities, just a whole lot messier
than neglecting that linear term.
